Identification of the dynamic characteristics of nonlinear structures

 
 
(3-73)
n = O
and since, from the orthogonality relationship of 
orthogonal to 
which is a homogeneous
therefore equation (3-73) can be rewritten as
the 
for 
are
functional of second degree,
 
(3-74)
n = O
For n = 0, the average involving 
is:
G o
= 
= 
( 3 - 7 5 )
The average for n=l is:
= 0
(3-76)

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Identification of Nonlinearity Using Higher-order 
9 5
(3-72)
Equations 
serve as a basis for the measurement of Wiener kernels. To
illustrate the derivation of these equations, consider the calculation of the second-order
kernel 
From equation 
becomes
 
 
(3-73)
n = O
and since, from the orthogonality relationship of 
orthogonal to 
which is a homogeneous
therefore equation 
(3-73) can 
be rewritten as
the 
for 
are
functional of second degree,
 
 
(3-74)
n = O
For n = 0, the average involving 
is:
 
 
 
 
 = 
 
 
 = 
( 3 - 7 5 )
The 
average for n=l is:
= 0
(3-76)

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Identification of Nonlinearity Using Higher-or&r 
9 6
since the average of the product of an odd number of zero-mean Gaussian variables is
zero. Finally the average for n = 2 is:
 
J J
J
J J
[ 
+ 
+ 
J =
J
J
= 
(3-77)
Combining equations 
equation (3-71) is obtained.
From the process of the above derivation, it can be seen that care must be taken in
applying the Wiener theory to practical problems because it is strictly valid only when the
averages of infinite time series are considered and Gaussian white noise input is assumed.
What can be obtained in practical calculation is an estimate of the true Wiener kernel and
the accuracy of the estimation depends on the length of the averaging time, the

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3
Identification of Nonlinearity Using Higher-order 
9 7
characteristics of the system 
LO 
be investigated and the closeness of the input signal to the
white Gaussian noise process. These points will be discussed further later on.
If the 
Wiener kernel
has been measured, then its corresponding
frequency response function 
is defined as
(3-78)
where 
is the 
Wiener kernel transform of 
As
discussed, when the input is low (the power spectrum of input A+ 0), the measured
Wiener kernels approach their corresponding Volterra kernels and, therefore, the
measured 
 
based on (3-78) approaches the Volterra kernel transform
3.5 MEASUREMENT OF WIENER KERNEL TRANSFORMS
3.5.1 MEASUREMENT OF WIENER KERNELS USING CORRELATION
ANALYSIS WITH RANDOM INPUT
So far, the theoretical basis for Wiener kernel measurement has been introduced and the
possibility of measuring these kernels from practical nonlinear structures now needs to be
assessed. Such an assessment can be carried out by simulating the measurement of
Wiener kernels of realistic nonlinear mechanical systems. The input random signal is a
band-limited Gaussian noise (the effect of non-white Gaussian noise input on the
estimation of Wiener kernels is discussed in 
since according to the sampling
theorem 
the maximum valid frequency 
(the Nyquist frequency) is limited by the
sampling rate l/At and a true white Gaussian noise signal is therefore impossible to
achieve. The process of generating band-limited Gaussian noise is done by passing the
sampled standard white Gaussian noise data with sampling frequency at l/At through a
band-limited filter to remove the higher frequency components. The numerical realisation
of this process is to interpolate the standard Gaussian noise data by a smoothing function
which, in the frequency domain, is an ideal low-pass filter 
The time
history and its power spectrum of one of the input band-limited noise signals with
sampling frequency at 
are shown in Fig.3.15.

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